This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Here are the solutions to the questions: 6. Solve the following question a) Convert 500 cm to m. Step 1: Recall the conversion factor between centimeters and meters. 1 m = 100 cm Step 2: Convert 500 cm to meters. 500 cm = (500)/(100) m = 5 m The conversion is 5 m. b) If the perimeter and length of the rectangle are 20 cm and 6 cm respectively, calculate the breadth of the rectangle. Step 1: Write the formula for the perimeter of a rectangle. P = 2(l + b) where P is the perimeter, l is the length, and b is the breadth. Step 2: Substitute the given values (P = 20 cm, l = 6 cm) into the formula. 20 = 2(6 + b) Step 3: Solve for b. (20)/(2) = 6 + b 10 = 6 + b b = 10 - 6 b = 4 cm The breadth of the rectangle is 4 cm. c) Find the volume of the given cuboid. Step 1: Identify the dimensions of the cuboid from the image. Length (l) = 5 cm Width (w) = 3 cm Height (h) = 4 cm Step 2: Write the formula for the volume of a cuboid. V = l × w × h Step 3: Substitute the dimensions and calculate the volume. V = 5 cm × 3 cm × 4 cm V = 15 cm^2 × 4 cm V = 60 cm^3 The volume of the cuboid is 60 cm^3. 7. Solve the following question a) Write in indices: 3^2 × 6 × 3. Step 1: Express all numbers as products of their prime factors. 6 = 2 × 3 Step 2: Substitute the prime factors back into the expression. 3^2 × (2 × 3) × 3 Step 3: Group terms with the same base and apply the exponent rule a^m × a^n = a^m+n. 2 × 3^2 × 3^1 × 3^1 2 × 3^(2+1+1) 2 × 3^4 The expression in indices is 2 × 3^4. b) Simplify: 8a + 6b - 4a. Step 1: Identify and group like terms. (8a - 4a) + 6b Step 2: Combine the like terms. 4a + 6b The simplified expression is 4a + 6b. 8. Solve the following question a) Multiply: 5x × x - 5x. Step 1: Perform the multiplication first. 5x × x = 5x^2 Step 2: Substitute the result back into the expression. 5x^2 - 5x The multiplied expression is 5x^2 - 5x. b) Divide: 20x^3 ÷ 4x^2y. Step 1: Write the division as a fraction. (20x^3)/(4x^2y) Step 2: Divide the coefficients and apply the exponent rule (a^m)/(a^n) = a^m-n for variables. (20)/(4) × (x^3)/(x^2) × (1)/(y) 5 × x^(3-2) × (1)/(y) 5x × (1)/(y) (5x)/(y) The result of the division is (5x)/(y). c) 1 is a prime number. Is this statement true, false or open? Step 1: Recall the definition of a prime number. A prime number is a natural number greater than 1 that has exactly two distinct positive divisors: 1 and itself. Step 2: Evaluate the number 1 against this definition. The number 1 is not greater than 1, and it only has one positive divisor (which is 1). Therefore, 1 does not fit the definition of a prime number. The statement "1 is a prime number" is False.